Global Sufficient Optimality Conditions for a Special Cubic Minimization Problem

Xiaomei Zhang,Weimin Ma,Yanjun Wang

doi:10.1155/2012/871741

Xiaomei Zhang, Weimin Ma + Show 1 more

Open Access

https://doi.org/10.1155/2012/871741

Copy DOI

Abstract

We present some sufficient global optimality conditions for a special cubic minimization problem with box constraints or binary constraints by extending the global subdifferential approach proposed by V. Jeyakumar et al. (2006). The present conditions generalize the results developed in the work of V. Jeyakumar et al. where a quadratic minimization problem with box constraints or binary constraints was considered. In addition, a special diagonal matrix is constructed, which is used to provide a convenient method for justifying the proposed sufficient conditions. Then, the reformulation of the sufficient conditions follows. It is worth noting that this reformulation is also applicable to the quadratic minimization problem with box or binary constraints considered in the works of V. Jeyakumar et al. (2006) and Y. Wang et al. (2010). Finally some examples demonstrate that our optimality conditions can effectively be used for identifying global minimizers of the certain nonconvex cubic minimization problem.

Highlights

Consider the following cubic minimization problem with box constraints: min f x bT x3 1 xT Ax aT x, 2 n s.t. x ∈ D ui, vi, i1CP 1 where ui, vi ∈ R, ui ≤ vi, i 1, 2, . . . , n, and a a1, . . . , an T ∈ Rn, b b1, . . . , bn T ∈ Rn, A ∈ Sn, where Sn is the set of all symmetric n × n matrices. x3 that means x13, . . . , xn[3] T .Mathematical Problems in EngineeringThe cubic optimization problem has spawned a variety of applications, especially in cubic polynomial approximation optimization 1, convex optimization 2, engineering design, and structural optimization 3
We present some sufficient global optimality conditions for a special cubic minimization problem with box constraints or binary constraints by extending the global subdifferential approach proposed by V
Research results about cubic optimization problem can be applied to quadratic programming problems, which have been widely studied because of their broad applications, to enrich quadratic programming theory

Summary

Introduction

Consider the following cubic minimization problem with box constraints: min f x bT x3 1 xT Ax aT x, 2 n s.t. x ∈ D ui, vi , i1. Several general approaches can be used to establish optimality conditions for solutions to optimization problems These approaches can be broadly classified into three groups: convex duality theory 4 , local subdifferentials by linear functions 5–7 , and global Lsubdifferential and L-normal cone by quadratic functions 8–11. We show how an L-subdifferential can be explicitly calculated for cubic functions and develop the global sufficient optimality conditions for CP 1. We derive the global optimality conditions for special cubic minimization problems with binary constraints. We rewrite the sufficient conditions in an other way through constructing a certain diagonal matrix This method is applicable to the quadratic minimization problem with box or binary constraints considered in 8, 12.

L-Subdifferentials and Sufficient Conditions

Sufficient Conditions of Bivalent Programming

Numerical Examples